Two lower bound arguments with "inaccessible" numbers

We present lower bound arguments for two general computational models: linear decision trees (LDT's) and random access machines (RAM's). Both proofs use (besides combinatorial and geometrical arguments) the method of constructing "hard" instances (x1, ..., xn) of the considered problems, where the distances between some of the xi are chosen so large that from the point of view of a fixed computational model the larger numbers are "inaccessible" from the smaller ones. In §2 we further refine this technique: there we have to satisfy at the same time equalities between certain sums of input numbers in order to allow a "fooling argument". The mentioned techniques allow us to derive sharper lower bounds for a variety of computational problems, including KNAPSACK, SHORTEST PATH and ELEMENT DISTINCTNESS.